124 Mr. G. F. C. Searle on the 



In practical cases, however, the electrification is not con- 

 centrated in ideal point-charges, but is spread over surfaces 

 or distributed through volumes. We therefore take a second 

 plane GG', parallel to FF' and at a distance dz from it, and a 

 corresponding point g on OP. Then, at the time t, the only 

 points on OP, which experience any wave disturbance from 

 the electrification dQ contained between the planes FF X and 

 GG', are the points between /and g, and, further, the wave 

 disturbance at points between f and g at time t is entirely 

 due to the electrification dQ. It is clear that, in the limit, 

 the electric force in the pulse at any point between / and g 

 may be calculated by substituting <^Q for q in (7) and then 

 supposing that de, the corresponding total flux of electric 

 displacement, no longer takes place along a mathematical 

 surface, but is uniformly distributed in a layer of thickness 

 dz. We thus obtain for the tangential electric force at any 

 point of the pulse, 



E = v S1U ° d< ^ (ID 



Kr(v — u cos 6) dz ^ ' 



In the same way, the corresponding total flux of magnetic 

 induction, fidh, is uniformly distributed through a layer of 

 thickness dz, and thus, by (10), the magnetic force in the 

 pulse is given by 



H = P KE = --/^^ Q . . . , . (12) 

 r(v — u cos 6)dz 



Since fjuKv 2 ='k, it follows that, at any point of the pulse, 

 j itH 2 /87r = / L6r 2 K 2 E 2 /87r = KE 2 /87r, and hence the magnetic 

 energy per unit volume is equal to the electric energy per 

 unit volume. Thus, when dQ/dz is known in terms of the 

 lingular coordinates of P, we can calculate the energy in 

 the pulse due to the impulsive change in the velocity of the 

 system. 



§ 8. Within the region of wave disturbance there is a 

 gradual transition of the field just outside the expanding 

 pulse into the field just inside the pulse. In each of these 

 fields the electric force ultimately varies as 1/r 2 , whereas the 

 tangential electric force, arising; from the sudden change in 

 the velocity of the system, ultimately varies as 1/r. Hence, 

 when r becomes very great, the only electric force which 

 need be considered is the tangential electric force E, given 



by (11). 



It we wish to do so, we can easily calculate the actual 

 electric and magnetic forces at any point/ (fig. 2) within the 

 pulse at any great time after the body has been set into 



